Animated Solution for Physics - Electrostatics: A metallic solid sphere is placed in a uniform electric field. The lines of force follow the path(s) shown in figure as

A Sphere in a Field

Imagine a perfectly smooth, solid metallic sphere resting quietly in a region of space. Suddenly, a uniform electric field is switched on. The field lines, like invisible arrows of force, march uniformly toward the sphere. But what happens when they encounter this metallic obstacle? Do they pierce straight through it like light through glass, or do they alter their path?

This classic problem tests our fundamental understanding of how conductors behave under electrostatic conditions. We are presented with four potential paths, and our mission is to deduce the only physically possible reality.

The Shielded Interior

To solve this, we must invoke the first great commandment of electrostatics for conductors: The electric field inside a conductor in electrostatic equilibrium is always exactly zero.

Why does this happen? A metal is teeming with a vast ocean of free electrons. The moment an external electric field is applied, these electrons experience a force and rush to the surface. They redistribute themselves in a fraction of a nanosecond, creating their own internal electric field that perfectly cancels out the external field.

Because the net electric field inside the metal is zero ($E_{\text{inside}}=0$), no electric field lines can exist inside the sphere.

Looking at our suspects, paths 1, 2, and 3 all depict field lines brazenly passing through the interior of the sphere. Based on our first law, we can immediately eliminate these three paths as physical impossibilities.

The Perpendicular Strike

Now we turn our attention to the surface of the sphere. The redistributed charges on the surface of the conductor ensure that the entire surface is at the exact same electrical potential. In other words, the surface of a conductor is an equipotential surface.

There is a beautiful geometric relationship between electric field lines and equipotential surfaces: they must always intersect at exactly $90^{\circ }$. If an electric field line hit the surface at an angle, it would have a tangential component. This tangential field would exert a force on the surface electrons, causing them to move. But we are in electrostatic equilibrium—nothing is moving! Therefore, the tangential component must be zero, meaning the field lines must strike the surface perfectly perpendicularly (normally).

Let's examine path 4. As the field line approaches the sphere, it gracefully bends. It doesn't just hit the sphere randomly; it curves specifically so that when it makes contact, it strikes the surface at a perfect $90^{\circ }$ angle.

Once it hits the surface, it terminates (on an induced negative charge). The interior of the sphere remains a pristine, field-free void. On the opposite side, a new field line originates (from an induced positive charge), emerging perfectly perpendicularly before bending back to join the uniform flow.

Path 4 flawlessly obeys both the law of the shielded interior and the law of the perpendicular strike. It is the only correct representation of the physics at play.